Probability and Probability Distributions

1

• Chapter 4
• Probability and Probability Distributions

2
What is Probability?

• In Chapters 2 and 3, we used graphs and numerical
  measures to describe data sets which were usually
  samples.
• We measured how often using

Relative frequency f/n

• As n gets larger,

Population
Probability
3
Basic Concepts

• An experiment is the process by which an
  observation (or measurement) is obtained.
• Experiment Recording an age
• Experiment Tossing a die
• Experiment Recording an opinion(yes, no)
• Experiment Tossing two coins

4
Basic Concepts

• A simple event is the outcome that is observed on
  a single repetition of the experiment.
• The basic element to which probability is
  applied.
• One and only one simple event can occur when the
  experiment is performed.
• A simple event is denoted by E with a subscript.
  Ei

5
Example The die toss

• Simple events

E1 E2 E3 E4 E5 E6
6
Basic Concepts

• Each simple event will be assigned a probability,
  measuring how often it occurs.
• The set of all simple events of an experiment is
  called the sample space, denoted by S.

7
Example The die toss

• Simple events Sample space

E1 E2 E3 E4 E5 E6
S E1, E2, E3, E4, E5, E6
8
Basic Concepts

• An event is a collection of one or more simple
  events.

• E1

• E3

• The die toss
• A an odd number
• B a number gt 2

• E5

• E2

• E6

• E4

A E1, E3, E5
B E3, E4, E5, E6
9
Basic Concepts

• Two events are mutually exclusive if, when one
  event occurs, the other cannot occur, and vice
  versa.

• Experiment Toss a die
• A observe an odd number
• B observe a number greater than 2
• C observe a 6
• D observe a 3

Not Mutually Exclusive
Mutually Exclusive
B and C? B and D?
10
The Probability of an Event

• The probability of an event A measures how
  often the event A will occur. We write P(A).
• Suppose that an experiment is performed n times.
  The relative frequency for an event A is

• If we let n get infinitely large,

11
Finding Probabilities

• Probabilities of simple event can be found by
  using
• Estimates from empirical studies
• Common sense estimates based on equally likely
  events.

• Examples
• Toss a fair coin.

P(Head) 1/2

• We know that 10 of the U.S. population has red
  hair. Now select a person at random.

P(Red hair) .10
12
The Probability of an Event

• P(A) must be between 0 and 1.
• If event A can never occur, then P(A) 0.
• If event A always occurs when the experiment is
  performed, then P(A) 1.
• The sum of the probabilities for all simple
  events in S equals 1.

• The probability of an event A is found by adding
  the probabilities of all the simple events
  contained in A.

13
Example

• Toss a fair coin twice. What is the probability
  of observing at least one head?

P(Ei)
1/4 1/4 1/4 1/4
P(at least 1 head) P(E1) P(E2) P(E3)
1/4 1/4 1/4 3/4
14
Example

• A bowl contains three MMs, one red, one blue
  and one green. A child selects two MMs at
  random. What is the probability that at least one
  is red?

1st 2nd Ei P(Ei)
1/6 1/6 1/6 1/6 1/6 1/6
P(at least 1 red) P(RB) P(BR) P(RG)
P(GR) 4/6 2/3
15
Counting Rules

• If the simple events in an experiment are equally
  likely, you can calculate

• You can use counting rules to find nA and N.

16
The mn Rule

• If an experiment is performed in two stages, with
  m ways to accomplish the first stage and n ways
  to accomplish the second stage, then there are mn
  ways to accomplish the experiment.
• This rule can be easily extended to k stages,
  with the number of ways equal to
• n1 n2 n3 nk

Example Toss two coins. The total number of
simple events is
2 ? 2 4
17
Examples
Example Toss three coins. The total number of
simple events is
2 ? 2 ? 2 8
Example Toss two dice. The total number of
simple events is
6 ? 6 36
Example Two MMs are drawn from a dish
containing two red and two blue candies. The
total number of simple events is
4 ? 3 12
18
Permutations

• Assume that there are n distinct objects, from
  which you take r objects at a time and arrange
  them in order. The number of different ways you
  can take and arrange is

Note that the order is important in a permutation.
19
Example
Choose three different digits from the numbers 1,
2, 3, and 4 as the key to 3-digit lock. How many
combinations can we make?
20
Examples
Example A toy consists of five parts and can be
assembled in any order. A child wants to try
different order of assembly. How many orders are
there?
21
Combinations

• Assume that there are n distinct objects, from
  which you select r objects at a time without
  regard to the order. The number of different ways
  you can select is

Note that the order is irrelevant in a
combination.
22
Example
Three members of a 5-person committee must be
chosen to form a subcommittee. How many different
subcommittees could be formed?
23
Example

• A box contains six MMs, four red and two green.
  A child selects two MMs at random. What is the
  probability that exactly one is red?

The order of the choice is not important!
4 ? 2 8 ways to choose 1 red and 1 green MM.
P(exactly one red) 8/15
24
In-Class Exercise

• In how many ways can you select five people from
  a group of eight if order of selection is
  important?
• In how many ways can you select two people from a
  group of 20 if order of selection is NOT
  important?

25
In-Class Exercise

• A sample space contains 10 simple events
  E1,E2,E10. If P(E1)3P(E2)0.45 and the
  remaining events have the same probability, find
  the probability of these remaining simple events.

26
In-Class Exercise

• Five cards are selected from a 52-card deck for a
  poker hand.
• How many possible poker hands can be dealt?
• In how many ways can you receive four cards of
  the same face value and one card from the other
  48 available?
• What is the probability of being dealt four of a
  kind?

27
Event Relations

• The union of two events, A and B, is the event
  which occurs if either A or B or both occur. We
  write
• A ??B

28
Event Relations

• The intersection of two events, A and B, is the
  event which occurs if both A and B occur. We
  write A?B.

• If two events A and B are mutually exclusive,
  then P(A ??B) 0.

29
Event Relations

• The complement of an event A is the event which
  occurs if event A does not occur. We write AC.

AC
AC consists of all outcomes of the experiment
that are not in A.
30
Example

• Select a student from the classroom and
• record his/her hair color and gender.
• A student has brown hair
• B student is female
• C student is male

• What is the relationship between events B and C?
• AC
• B?C
• B?C

Not only mutually exclusive but also B CC
Student does not have brown hair
Student is both male and female ?
Student is either male and female all students
S
31
Calculating Probabilities for Unions and
Complements

• There are special rules that will allow you to
  calculate probabilities for composite events.
• The Additive Rule for Unions
• For any two events, A and B, the probability of
  their union, P(A ??B), is

32
Example Additive Rule
Example Suppose that there were 120 students in
the classroom, and that they could be classified
as follows
A brown hair P(A) 50/120 B female P(B)
60/120
P(A?B) P(A) P(B) P(A?B) 50/120 60/120 -
30/120 80/120 2/3
Check P(A?B) (20 30 30)/1202/3
33
A Special Case
When two events A and B are mutually exclusive,
P(A?B) 0 and P(A?B) P(A) P(B).
A male with brown hair P(A) 20/120 B female
with brown hair P(B) 30/120
P(A?B) P(A) P(B) 20/120 30/120 50/120
34
Calculating Probabilities for Complements

• We know that for any event A
• P(A ??AC) 0
• Since either A or AC must occur,
• P(A ??AC) 1
• so that P(A ??AC) P(A) P(AC) 1

P(AC) 1 P(A)
35
Example
Select a student at random from the classroom.
A male P(A) 60/120 B female
P(B) 1- P(A) 1- 60/120 60/120
36
Calculating Probabilities for Unions
37
Example Additive Rule
Example Suppose that there were 120 students in
the classroom, and that they could be classified
as follows
A brown hair P(A) 50/120 B female P(B)
60/120
P(A?B) P(A) P(B) P(A?B) 50/120 60/120 -
30/120 80/120 2/3
38
Calculating Probabilities for Intersections

• In the previous example, we found P(A ? B)
  directly from the table. Sometimes this is
  impractical or impossible.
• The rule for calculating P(A ? B) depends on the
  idea of independent and dependent events.

39
Two events, A and B, are said to be independent
if and only if the probability that event A
occurs is not influenced or changed by the
occurrence of event B , or vice versa.
Example Toss a fair coin twice.

• A head appears on second toss
• B head appears on first toss

A and B are independent.
40
Example

• A bag contains two MMs, one red and one blue.
  Let us take two candies out one by one, and
  define
• A second candy is red.
• B first candy is blue.

A and B are dependent,
because probability that event A occurs is
influenced by the occurrence of event B.
41
Conditional Probabilities

• The probability that A occurs, given that event
  B has occurred, is called the conditional
  probability of A given B, which is defined as

42
Example 1

• Toss a fair coin twice.
• A head on second toss
• B head on first toss

P(AB) 1/2 P(Anot B) 1/2
Probability that A occurs does not change,
whether B happens or not
43
Example 2

• A bowl contains five MMs, two red and three
  blue. Randomly select two candies, and define
• A second candy is red.
• B first candy is blue.

P(AB) P(2nd red1st blue) 2/4 1/2 P(Anot B)
P(2nd red1st red) 1/4
Probability that A occurs does change, depending
on whether B happens or not
44
Defining Independence

• We can redefine independence in terms of
  conditional probabilities

Two events A and B are independent if and only
if P(AB) P(A) or P(BA) P(B) Otherwise,
they are dependent.

• Once youve decided whether or not two events are
  independent, then you can use the following rule
  to calculate their intersection.

45
The Multiplicative Rule for Intersections

• For any two events, A and B, the probability that
  both A and B occur is

P(A ??B) P(A)P(BA)

• If the events A and B are independent, then the
  probability that both A and B occur is

P(A ??B) P(A) P(B)
46
Example 1
In a large population of patients, 10 of the
patients can be classified as being high risk for
a heart attack. Three patients are randomly
selected from this population. What is the
probability that exactly one of the three are
high risk?
H high risk N not high risk.
P(exactly one high risk) P(HNN) P(NHN)
P(NNH) P(H)P(N)P(N) P(N)P(H)P(N)
P(N)P(N)P(H) (.1)(.9)(.9) (.9)(.1)(.9)
(.9)(.9)(.1) 3(.1)(.9)2 .243
47
Example 2
Suppose we have additional information in the
previous example. We know that only 49 of the
patients are female. Also, of the female
patients, 8 are high risk. A single person is
selected at random. What is the probability that
it is a high risk female?
H high risk F female
From the example, P(F) .49 and P(HF) .08.
Use the Multiplicative Rule P(high risk female)
P(H?F) P(F)P(HF) .49(.08) .0392
48
Review

• The probability that A occurs, given that B has
  occurred, is called the conditional probability
  of A given B.

49
Review
Two events, A and B, are said to be independent
if and only if the probability that A occurs is
not influenced or changed by the occurrence of B
, or vice versa.
50
Example

• In the general population, the proportions of
  colorblind men and women are shown below

One person is drawn at random from the
population. Define A the person is colorblind
B the person is found to be a man.
Whether A and B are independent or not?
51
In-Class Exercise

• Toss a single die and observe the number of dots.
  Define
• Event A the number is less than 4
• Event B the number is less than or equal to 2
• Event C the number is greater than 3
• Are A and B independent? Mutually exclusive?
• Are A and C independent ? Mutually exclusive?

52
Review

• For any two events, A and B, the probability that
  both A and B occur is

P(A ??B) P(A)P(BA)

• If the events A and B are independent, then the
  probability that both A and B occur is

P(A ??B) P(A) P(B)
53
Example

• Two cards are drawn from a deck 52 cards.
  Calculate the probability that the draw includes
  an ace and a ten.

54
Random Variables

• A quantitative variable x is a random variable if
  the value that it assumes, corresponding to the
  outcome of an experiment, is a random event.
• Random variables can be discrete or continuous.

• Examples
• x SAT score for a randomly selected student
• x number of people in a supermarket at a
  randomly selected time of day
• x number on the upper face of a tossed die

55
Probability Distributions for Discrete Random
Variables

• The probability distribution for a discrete
  random variable x resembles the relative
  frequency distributions we constructed in Chapter
  1. It is a graph, table or formula that gives the
  possible values of x and the probability p(x)
  associated with each value.

56
Example

• Toss a fair coin three times and define x
  number of heads.

57
x 3 2 2 2 1 1 1 0
HHH
1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8
P(x 0) 1/8 P(x 1) 3/8 P(x 2)
3/8 P(x 3) 1/8
HHT
HTH
THH
HTT
THT
TTH
TTT
58
Probability Distributions

• Probability distributions can be used to describe
  the population, just as we described samples in
  Chapter 1.
• Shape Symmetric, skewed, mound-shaped
• Outliers unusual or unlikely measurements
• Center and spread mean and standard deviation.
  A population mean is called µ and a population
  standard deviation is called s.

59
The Mean and Standard Deviation

• Let x be a discrete random variable with
  probability distribution p(x). Then the mean,
  variance and standard deviation of x are given as

The population mean, which measures the average
value of x in the population, is also called the
expected value of x, and denoted by E(x)µ.
60
Example

• Toss a fair coin 3 times and record the number of
  heads, x .

61
Example

• The probability distribution for x the number of
  heads in tossing 3 fair coins.

Symmetric mound-shaped

• Shape?
• Outliers?
• Center?
• Spread?

None
m 1.5
s .688
62
Example

• In a lottery, 8,000 tickets are to be sold at 5
• each. The prize is a 12,000 automobile. If you
• purchased two tickets, what is your expected
• gain?

Define x your gain.
x -10 or 11,990
µ E(x) S xp(x) (-10)
(7998/8000)(11,990)(2/8000) -7
63
Key Concepts

• I. Experiments and the Sample Space
• Experiments, events, mutually exclusive events,
  simple events
• The sample space
• Venn diagrams, tree diagrams, probability tables

64
Key Concepts

• II. Probabilities
• 1. Relative frequency definition of probability
• 2. Properties of probabilities
• a. Each probability lies between 0 and 1.
• b. Sum of all simple-event probabilities
  
• equals1.
• 3. P(A), the sum of the probabilities for all
  simple events in A

65
Key Concepts

• III. Counting Rules
• 1. mn Rule extended mn Rule
• 2. Permutations
• 3. Combinations

66
Key Concepts

• IV. Event Relations
• 1. Unions and intersections
• 2. Events
• a. Disjoint or mutually exclusive
• P(A Ç B) 0
• b. Complementary P(A) 1 - P(AC )

67
Key Concepts

• 3. Conditional probability
• 4. Independent and dependent events
• 5. Additive Rule of Probability
• 6. Multiplicative Rule of Probability
• 7. Law of Total Probability
• 8. Bayes Rule

68
Key Concepts

• V. Discrete Random Variables and Probability
  Distributions
• 1. Random variables, discrete and continuous
• 2. Properties of probability distributions

69
Key Concepts

• 3. Mean or expected value of a discrete
• random variable
• 4. Variance and standard deviation of a
  discrete random variable